Local discontinuous Galerkin method for nonlinear BSPDEs of Neumann boundary conditions with deep backward dynamic programming time-marching

TL;DR

This paper introduces a local discontinuous Galerkin method combined with deep learning to efficiently solve nonlinear backward stochastic PDEs with Neumann boundary conditions, demonstrating stability, accuracy, and effectiveness.

Contribution

The paper develops a novel LDG scheme for BSPDEs with Neumann conditions and integrates deep learning to overcome high-dimensional challenges.

Findings

Proves $L^2$-stability and optimal error estimates for the scheme.

Demonstrates effectiveness through numerical examples.

Shows the method's accuracy in high-dimensional problems.

Abstract

This paper aims to present a local discontinuous Galerkin (LDG) method for solving backward stochastic partial differential equations (BSPDEs) with Neumann boundary conditions. We establish the $L^2$-stability and optimal error estimates of the proposed numerical scheme. Two numerical examples are provided to demonstrate the performance of the LDG method, where we incorporate a deep learning algorithm to address the challenge of the curse of dimensionality in backward stochastic differential equations (BSDEs). The results show the effectiveness and accuracy of the LDG method in tackling BSPDEs with Neumann boundary conditions.